Abstract

By utilizing the novel coordinate system for Gaussian beam reflection and the generalized ray matrix for spherical mirror reflection, the generalized sensitivity factors SD1, ST1, SD2 and ST2 influenced by both the radial and axial displacements of a spherical mirror in a nonplanar ring resonator have been obtained. Besides, the singular points of different kinds of non-planar ring resonators under the conditions of incident angle A ranging from 0° to 45° or total coordinate rotation angle ρ ranging from 0°to 360°have also been obtained through the analysis of the determinant of the coefficient matrix of the linear equations. The analysis in this paper is important to the cavity design of non-planar ring resonators and it could be helpful to avoid the violent movement of the optical-axis to small misalignment of the mirrors in non-planar ring resonators.

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  1. W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
    [CrossRef]
  2. A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron.6(6), 1389–1399 (2000).
    [CrossRef]
  3. A. E. Siegman, Lasers (University Science, 1986).
  4. G. A. Massey and A. E. Siegman, “Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces,” Appl. Opt.8(5), 975–978 (1969).
    [CrossRef] [PubMed]
  5. H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, “The multioscillator ring laser gyroscope,” in Laser Handbook, M. I. Stitch, and M. Bass, eds. 4, Chap. 3, 229–327, (North-Holland, 1985).
  6. S.-C. Sheng, “Optical-axis perturbation singularity in an out-of-plane ring resonator,” Opt. Lett.19(10), 683–685 (1994).
    [CrossRef] [PubMed]
  7. J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun.281(5), 1204–1210 (2008).
    [CrossRef]
  8. D. D. Wen, D. Li, and J. L. Zhao, “Generalized sensitivity factors for optical-axis perturbation in nonplanar ring resonators,” Opt. Express19(20), 19752–19757 (2011).
    [CrossRef] [PubMed]
  9. J. Yuan, M. Chen, Z. Kang, and X. Long, “Novel coordinate system for Gaussian beam reflection,” Opt. Lett.37(11), 2082–2084 (2012).
    [CrossRef] [PubMed]
  10. J. Yuan, X. W. Long, and M. X. Chen, “Generalized ray matrix for spherical mirror reflection and its application in square ring resonators and monolithic triaxial ring resonators,” Opt. Express19(7), 6762–6776 (2011).
    [CrossRef] [PubMed]
  11. J. Yuan, X. Long, L. Liang, B. Zhang, F. Wang, and H. Zhao, “Nonplanar ring resonator modes: generalized Gaussian beams,” Appl. Opt.46(15), 2980–2989 (2007).
    [CrossRef] [PubMed]
  12. G. J. Martin, “Multioscillator ring laser gyro using compensated optical wedge,” U.S. patent 5,907,402 (25 May 1999).
  13. J. Yuan, X. W. Long, B. Zhang, F. Wang, and H. C. Zhao, “Optical axis perturbation in folded planar ring resonators,” Appl. Opt.46(25), 6314–6322 (2007).
    [CrossRef] [PubMed]
  14. A. H. Paxton and W. P. Latham., “Unstable resonators with 90 ° beam rotation,” Appl. Opt.25(17), 2939–2946 (1986).
    [CrossRef] [PubMed]
  15. J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt.47(5), 628–631 (2008).
    [CrossRef] [PubMed]
  16. X. W. Long and J. Yuan, “Method for eliminating mismatching error in monolithic triaxial ring resonators,” Chin. Opt. Lett.8(12), 1135–1138 (2010).
    [CrossRef]
  17. Y. X. Zhao, M. G. Sceats, and A. D. Stokes, “Application of ray tracing to the design of a monolithic nonplanar ring laser,” Appl. Opt.30(36), 5235–5238 (1991).
    [CrossRef] [PubMed]
  18. H. T. Tuan and S. L. Huang, “Analysis of reentrant two-mirror nonplanar ring laser cavity,” J. Opt. Soc. Am. A22(11), 2476–2482 (2005).
    [CrossRef] [PubMed]
  19. S. Gangopadhyay and S. Sarkar, “ABCD matrix for reflection and refraction of Gaussian light beams at surfaces of hyperboloid of revolution and efficiency computation for laser diode to single-mode fiber coupling by way of a hyperbolic lens on the fiber tip,” Appl. Opt.36(33), 8582–8586 (1997).
    [CrossRef] [PubMed]
  20. H. Z. Liu, L. R. Liu, R. W. Xu, and Z. Luan, “ABCD matrix for reflection and refraction of Gaussian beams at the surface of a parabola of revolution,” Appl. Opt.44(23), 4809–4813 (2005).
    [CrossRef] [PubMed]

2012 (1)

2011 (2)

2010 (1)

2008 (2)

J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun.281(5), 1204–1210 (2008).
[CrossRef]

J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt.47(5), 628–631 (2008).
[CrossRef] [PubMed]

2007 (2)

2005 (2)

2000 (1)

A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron.6(6), 1389–1399 (2000).
[CrossRef]

1997 (1)

1994 (1)

1991 (1)

1986 (1)

1985 (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

1969 (1)

Chen, M.

Chen, M. X.

Chow, W. W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Gangopadhyay, S.

Gea-Banacloche, J.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Huang, S. L.

Kang, Z.

Latham, W. P.

Li, D.

Liang, L.

Liang, L. M.

Liu, H. Z.

Liu, L. R.

Long, X.

Long, X. W.

Luan, Z.

Massey, G. A.

Paxton, A. H.

Pedrotti, L. M.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Sanders, V. E.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Sarkar, S.

Sceats, M. G.

Schleich, W.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Scully, M. O.

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Sheng, S.-C.

Siegman, A. E.

A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron.6(6), 1389–1399 (2000).
[CrossRef]

G. A. Massey and A. E. Siegman, “Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces,” Appl. Opt.8(5), 975–978 (1969).
[CrossRef] [PubMed]

Stokes, A. D.

Tuan, H. T.

Wang, F.

Wen, D. D.

Xu, R. W.

Yuan, J.

Zhang, B.

Zhao, H.

Zhao, H. C.

Zhao, J. L.

Zhao, Y. X.

Appl. Opt. (8)

G. A. Massey and A. E. Siegman, “Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces,” Appl. Opt.8(5), 975–978 (1969).
[CrossRef] [PubMed]

J. Yuan, X. W. Long, B. Zhang, F. Wang, and H. C. Zhao, “Optical axis perturbation in folded planar ring resonators,” Appl. Opt.46(25), 6314–6322 (2007).
[CrossRef] [PubMed]

A. H. Paxton and W. P. Latham., “Unstable resonators with 90 ° beam rotation,” Appl. Opt.25(17), 2939–2946 (1986).
[CrossRef] [PubMed]

J. Yuan, X. W. Long, and L. M. Liang, “Optical-axis perturbation in triaxial ring resonator,” Appl. Opt.47(5), 628–631 (2008).
[CrossRef] [PubMed]

J. Yuan, X. Long, L. Liang, B. Zhang, F. Wang, and H. Zhao, “Nonplanar ring resonator modes: generalized Gaussian beams,” Appl. Opt.46(15), 2980–2989 (2007).
[CrossRef] [PubMed]

Y. X. Zhao, M. G. Sceats, and A. D. Stokes, “Application of ray tracing to the design of a monolithic nonplanar ring laser,” Appl. Opt.30(36), 5235–5238 (1991).
[CrossRef] [PubMed]

S. Gangopadhyay and S. Sarkar, “ABCD matrix for reflection and refraction of Gaussian light beams at surfaces of hyperboloid of revolution and efficiency computation for laser diode to single-mode fiber coupling by way of a hyperbolic lens on the fiber tip,” Appl. Opt.36(33), 8582–8586 (1997).
[CrossRef] [PubMed]

H. Z. Liu, L. R. Liu, R. W. Xu, and Z. Luan, “ABCD matrix for reflection and refraction of Gaussian beams at the surface of a parabola of revolution,” Appl. Opt.44(23), 4809–4813 (2005).
[CrossRef] [PubMed]

Chin. Opt. Lett. (1)

IEEE J. Sel. Top. Quantum Electron. (1)

A. E. Siegman, “Laser beams and resonators: beyond the 1960s,” IEEE J. Sel. Top. Quantum Electron.6(6), 1389–1399 (2000).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

J. Yuan and X. W. Long, “Optical-axis perturbation in nonplanar ring resonators,” Opt. Commun.281(5), 1204–1210 (2008).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Rev. Mod. Phys. (1)

W. W. Chow, J. Gea-Banacloche, L. M. Pedrotti, V. E. Sanders, W. Schleich, and M. O. Scully, “The ring laser gyro,” Rev. Mod. Phys.57(1), 61–104 (1985).
[CrossRef]

Other (3)

A. E. Siegman, Lasers (University Science, 1986).

H. Statz, T. A. Dorschner, M. Holtz, and I. W. Smith, “The multioscillator ring laser gyroscope,” in Laser Handbook, M. I. Stitch, and M. Bass, eds. 4, Chap. 3, 229–327, (North-Holland, 1985).

G. J. Martin, “Multioscillator ring laser gyro using compensated optical wedge,” U.S. patent 5,907,402 (25 May 1999).

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Figures (7)

Fig. 1
Fig. 1

Geometrical construction of a four-equal-sided non-planar ring resonator (NPRO), mj(j = 1,2,3,4):reflecting mirror with radius of Rj(j = 1,2,3,4), Pj(j = 1,2,3,4): terminal points of the resonator.

Fig. 2
Fig. 2

Coordinate systems and corresponding coordinate rotations based on traditional coordinate system for Gaussian beam reflection (TCS) and novel coordinate system for Gaussian beam reflection (NCS) in four equal-sided non-planar ring resonators (NPRO), β: folding angle, m1 and m2: spherical mirrors with radius of R1 and R2, m3 and m4: planar mirrors, Aj(j = 1,2,3,4): incident angles on four mirrors, Pj(j = 1,2,3,4): terminal points of the resonator, Pe, Pf, Pg, Ph, O1, O2: the midpoints of straight lines P1P2, P2P3, P3P4, P4P1, P1P3 and P2P4 separately, φtj(j = 1,2,3,4) and φj(j = 1,2,3,4): coordinate rotation angles based on TCS and NCS respectively, nj(j = 1,2,3,4): the binormals at points Pj(j = 1,2,3,4), (xtj, yj, zj) and (xj, yj, zj)(j = 1,2,3,4): coordinate systems for the incident beam (based on TCS and NCS respectively) before being reflected from points Pj(j = 1,2,3,4), (xtjr, yjr, zjr) and (xjr, yjr, zjr)(j = 1,2,3,4): coordinate systems for the reflected beam (based on TCS and NCS respectively) after being reflected from points Pj(j = 1,2,3,4), δjz(j = 1,2,3,4): axial displacement of mirrors mj(j = 1, 2, 3, 4), δjx, δjy(j = 1,2): radial displacements of the spherical mirrors m1 and m2. (Note: The positive directions of yj and yjr(j = 1,2,3,4) are along the directions of nj(j = 1,2,3,4); the positive directions of zj and zjr(j = 1,2,3,4,b,c) are along the direction of beam propagation; (xt1, xt1r, x1, x1r), (xt2, xt2r, x2, x2r), (xt3, xt3r, x3, x3r) and (xt4, xt4r, x4, x4r) are located at the incident planes of P4P1P2, P1P2P3, P2P3P4 and P3P4P1 separately; the positive directions of δ1x, δ2x, δ1z, δ2z, δ3z and δ4z are along the directions of straight lines P2P4, P1P3, P1O2, P2O1, P3O2 and P4O1 separately; the positive direction of δjy (j = 1,2) is along the direction of nj(j = 1,2).)

Fig. 3
Fig. 3

Incident angle A versus coordinate rotation angle φ

Fig. 4
Fig. 4

Sensitivity factors SD1 and ST1 characterizing the movement of the optical-axis on mirror m1 with A = 43.866°. The perturbation source is the angular misalignments of mirror m1.

Fig. 5
Fig. 5

Sensitivity factors SD2 and ST2 characterizing the movement of the optical-axis on mirror m1 with A = 43.866°. The perturbation source is the translational displacements of mirror m1.

Fig. 6
Fig. 6

Determinant of M' versues L/R.

Fig. 7
Fig. 7

Stability map of NPRO and the track of the singular points under the condition of (a) ρ ranging from 0°to 360°and L/R ranging from 0 to 2, (b) ρ ranging from 0°to 360°and L/R ranging from 2 to 8, (c) A ranging from 0°to 45°and L/R ranging from 0 to 2, (d) A ranging from 0°to 45°and L/R ranging from 2 to 8. (Note: the stable and unstable regions are separated with solid lines; the tracks of the singular points are illustrated with the red marked lines)

Equations (14)

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M( R j , A j )=[ 1 0 0 0 2 δ jz sin( A i ) 2 R j cos( A j ) 1 0 0 2 δ jz tan( A j ) R j +2( θ jx + δ jx R j ) 0 0 1 0 0 0 0 2×cos( A j ) R j 1 2( θ jy + δ jy R j ) 0 0 0 0 1 ].
M j =R( φ j )M( R j , A j )T( L j )
M j =R( φ j )M( m j )T( L j ).
M( m 1 )=[ 1 0 0 0 2 δ 1z sin(A) 2 Rcos(A) 1 0 0 2 δ 1z tan(A) R + 2 δ 1x R 0 0 1 0 0 0 0 2×cos(A) R 1 2 δ 1y R 0 0 0 0 1 ],
M( m 2 )=[ 1 0 0 0 2 δ 2z sin(A) 2 Rcos(A) 1 0 0 2 δ 2z tan(A) R + 2 δ 2x R 0 0 1 0 0 0 0 2×cos(A) R 1 2 δ 2y R 0 0 0 0 1 ],
M( m 3 )=[ 1 0 0 0 2 δ 3z sin(A) 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ],
M( m 4 )=[ 1 0 0 0 2 δ 4z sin(A) 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ].
R( φ j )=[ cos( φ j ) 0 sin( φ j ) 0 0 0 cos( φ j ) 0 sin( φ j ) 0 sin( φ j ) 0 cos( φ j ) 0 0 0 sin( φ j ) 0 cos( φ j ) 0 0 0 0 0 1 ]
φ t1 = φ t2 = φ t3 = φ t4 =φ,
φ 1 = φ 2 = φ 3 = φ 4 =φ.
sin (A) 2 = cos(φ) 1+cos(φ) .
ρ=| φ 1 |+| φ 2 |+| φ 3 |+| φ 4 |=4φ
M=R( φ 1 )M( m 1 )T( L 1 )R( φ 4 )M( m 4 )T( L 4 )R( φ 3 )M( m 3 )T( L 3 )R( φ 2 )M( m 2 )T( L 2 )
( r x r x ' r y r y ' 1 )=M( r x r x ' r y r y ' 1 ).

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